Duality on gradient estimates and Wasserstein controls
∗Kazumasa Kuwada†
(Graduate school of Humanities and Sciences, Ochanomizu university)
Let (X, d) be a Polish space. Let dand ˜d be continuous distance functions onX ( ˜dcan be different fromd). Forf :X→Rand x∈X, we define |∇df|(x) and k∇dfk∞ by
|∇df|(x) := lim
r↓0 sup
0<d(x,y)≤r
f(x)−f(y) d(x, y)
, k∇dfk∞:= sup
x∈X
|∇df|(x).
Note that k∇dfk∞ < ∞ holds if and only if f ∈ Lipd(X). Let P(X) be the space of all probability measures onX. Forp∈[1,∞] andµ, ν∈ P(X), we defineLp-Wasserstein (pseudo-) distance dpW(µ, ν) by
dpW(µ, ν) := inf
{kdkLp(π) π : coupling of µand ν }
. (I)
We define|∇d˜f|(x),k∇d˜fk∞ and ˜dpW similarly by using ˜dinstead ofd.
Let (Px)x∈X be a Markov kernel. Forp∈[1,∞], we consider the following two properties:
(i) (Lq-gradient estimate (of Bakry- ´Emery type)) For f ∈Lipd(X)∩Cb(X),
|∇d˜P f|(x)≤P(|∇df|q)(x)1/q (Gq) for anyx∈X when q <∞. When q=∞,
∇d˜P f
∞≤ k∇dfk∞. (G∞) (ii) (Lp-Wasserstein control) For anyµ, ν ∈ P(X),
dpW(P∗µ, P∗ν)≤d˜pW(µ, ν). (Cp) Assumption 1
(i) dand ˜dare compatible with the topology onX and (X, d) is locally compact.
(ii) For anyx, y ∈X there exists a curve γ : [0,1]→X fromx toy such thatd(γ(s), γ(t)) =
|s−t|d(x, y) for anys, t∈[0,1]. The same is true for ˜d.
(iii) There exists a positive Radon measurev on X such that
(a) (X, d, v) enjoys the local uniform volume doubling condition, that is, there are con- stantsD, R1>0 such thatv(B2r(x))≤Dv(Br(x)) holds for allx∈Xandr ∈(0, R1).
(b) (X, d, v) supports a (1, ρ)-local Poincar´e inequality for some ρ≥1, that is, for every R >0, there are constants λ≥1 and CP >0 such that, for anyf ∈Lipd(X),
∫
Br(x)
|f −fx,r|dv≤CPr {∫
Bλr(x)
|∇df|ρdv }1/ρ
(II) holds for everyx∈X andr ∈(0, R), wherefx,r :=v(Br(x))−1∫
Br(x)f dv.
∗研究集会「確率論シンポジウム」(2010年12月20日―12月23日 於数理解析研究所)講演予稿
†URL:http://www.math.ocha.ac.jp/kuwada e-mail:[email protected]
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(c) Px is absolutely continuous with respect tov for allx∈X;Px(dy) =Px(y)v(dy). In addition, the density Px(y) is continuous with respect tox.
Theorem 1 [3] Let p, q∈[1,∞] withp−1+q−1 = 1. Then the following hold.
(i) (Cp) implies (Gq) (without Assumption 1).
(ii) Suppose that Assumption 1 holds. Then (Gq) implies (Cp).
Example 1
(i) Let X be a Lie group of type H (e.g. a Heisenberg group), d the Carnot-Caratheodory distance and P = Pt the heat semigroup associated with the canonical sub-Laplacian.
Then (G1) holds with ˜d=Kdfor someK >1 (see [2]; K must be >1).
(ii) Let (X, d) be a Lie group with a bracket generating family{Xi}ni=1 of left-invariant vector fields, d the associated Carnot-Caratheodory distance and P = Pt the heat semigroup associated with ∑n
i=1Xi2. Then (Gq) holds for q > 1 with ˜d= Kq(t)d (see [5]). If X is nilpotent,Kq(t)≡Kq for some constantKq>1.
Assumption 1 is satisfied in the cases (i)(ii). Hence we can apply Theorem 1 to show (C∞) or (Cp) (1≤p <∞) respectively.
For the proof of Theorem 1 (ii), we use a general theory of Hamilton-Jacobi semigroup developed in [1, 4] in the context of abstract metric spaces. Given p∈[1,∞), we define Qtf by
Qtf(x) := inf
y∈X
[
f(y) +t·1 p
(d(x, y) t
)p]
forf ∈Cb(X). Under Assumption 1 (iii)(a) and (iii)(b), for t >0 and v-a.e. x∈X, lims↓0
Qt+sf(x)−Qtf(x)
s =−1
q|∇dQtf|(x)q.
The Kantorovich duality provides a expression of dpW(µ, ν) by means of the Hamilton-Jacobi semigroup and a sort of “the fundamental theorem of calculus” and “the Leibniz rule” implies the conclusion.
References
[1] Z.M. Balogh, A. Engoulatov, L. Hunziker, and O.E. Maasalo. Functional inequalities and Hamilton-Jacobi equations in geodesic spaces. preprint; arXiv:0906.0476.
[2] N. Eldredge. Gradient estimates for the subelliptic heat kernel on H-type groups. J. Funct.
Anal., 258(2):504–533, 2010.
[3] K. Kuwada. Duality on gradient estimates and Wasserstein controls. J. Funct. Anal., 258(11):3758–3774, 2010.
[4] J. Lott and C. Villani. Hamilton-Jacobi semigroup on length spaces and applications. J.
Math. Pures Appl. (9), 88(3):219–229, 2007.
[5] T. Melcher. Hypoelliptic heat kernel inequalities on Lie groups. Stochastic Process. Appl., 118(3):368–388, 2008.
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